# Geometric Series - absolutely convergent, conditionally convergent or divergent?

How would I determine if the following series is absolutely convergent, conditionally convergent or divergent?

$\sum\limits_{i=1}^n$$\sqrt[n]{2}+1$

• Firstly, this series is not geometric. Secondly, what have you tried? – James Nov 15 '15 at 2:07
• I tried to begin with the root test but got confused pretty quick – Glenn2329 Nov 15 '15 at 2:08
• Well the root test is just going to introduce another root, so, you will have a root of a root, which isn't really any simpler. The root test is good when you have $n$th powers, not $n$th roots. For a rough heuristic as to orders to test series convergence, I suggest doing the divergence test first i.e. does the *sequence* $(\sqrt[n]{2} + 1)$ tend to $0$ as $n\rightarrow\infty$? – James Nov 15 '15 at 2:11
• Do you mean $\sum_{n=1}^\infty(\sqrt[n]2+1)$? – Tim Raczkowski Nov 15 '15 at 2:24
• yes thats what i meant – Glenn2329 Nov 15 '15 at 2:31

Notice that $\lim_{n\rightarrow\infty}\sqrt[n]2+1=1+1=2\neq0$. Hence the series diverges.